Root Systems for Levi Factors and Borel–de Siebenthal Theory

Let m be a Levi factor of a proper parabolic subalgebra q of a complex semisimple Lie algebra g. Let t = centm. A nonzero element ν ∈ t is called a t-root if the corresponding adjoint weight space gν is not zero. If ν is a t-root, some time ago we proved that gν is adm irreducible. Based on this result we develop in the present paper a theory of t-roots which replicates much of the structure of classical root theory (case where t is a Cartan subalgebra). The results are applied to obtain new results about the structure of the nilradical n of q. Also applications in the case where dim t = 1 are used in Borel–de Siebenthal theory to determine irreducibility theorems for certain equal rank subalgebras of g. In fact the irreducibility results readily yield a proof of the main assertions of the Borel–de Siebenthal theory.